The interaction of a Hopf bifurcation with the codimension-2 cusp bifurcation yields a codimention three bifurcation with a rich variety of dynamic behaviour and new interesting phenamena which do not exist in the fold-Hopf case . This bifurcation is called codimension-3 cusp-Hopf bifurcation that is studied on the three dimensional center manifold . The focus of this thesis is determination of the dynamical behavior when a Hopf ifurcation occurs at an equilibrium point near the cusp bifurcation. O ur analysi is based o truncated normal form. The 1 symmetry of the normal form system leads to further reduction to a plannar vector field . Further transformations are used to simplify the nonlinear coefficient and reduse the number of cases under consideration . The invariant sets of the two dimensional truncatednormal form has been located. That includs all equilibrium points and period orbits . Codimension-1 bifurcations such as the secondary Hopf and pitchfork bifurcation are calculated in two dimensional state space . The periodic orbit bifurcated in the secondary Hopf bifurcation are associated with the torus bifurcation. The coalesense of three equilibria happe through pitchfork bifurcation. Bifurcation varieties are presented in three parameter space (? 1 , ? 2 , ? 3 ) which are called cusp , Hopf and torus varieties. When equilibriums point change from real to complex variable Then this varieties are created . Because of the transverse intersection of bifurcation varieties with any plane ? 3 = constant ? 0 , the study of behaviour of bifurcation varieties is reduced to three two—parameter cross sections , containing codimension--2 bifurcation points and codimension--1 bifurcation curves. In two of these bifurcation points , fold -- Hopf bifurcation occurs . Therefor, the cusp--Hopf bifurcation holds all four basic cases of the ltr"